Anonymity-Preserving Space Partitions

Abstract

We consider a multidimensional space partitioning problem, which we call Anonymity-Preserving Partition. Given a set P of n points in ?^d and a collection H of m axis-parallel hyperplanes, the hyperplanes of H partition the space into an arrangement A(H) of rectangular cells. Given an integer parameter t > 0, we call a cell C in this arrangement deficient if 0 < |C ? P| < t; that is, the cell contains at least one but fewer than t data points of P. Our problem is to remove the minimum number of hyperplanes from H so that there are no deficient cells. We show that the problem is NP-complete for all dimensions d ? 2. We present a polynomial-time d-approximation algorithm, for any fixed d, and we also show that the problem can be solved exactly in time (2d-0.924)^k m^O(1) + O(n), where k is the solution size. The one-dimensional case of the problem, where all hyperplanes are parallel, can be solved optimally in polynomial time, but we show that a related Interval Anonymity problem is NP-complete even in one dimension